Relate Mersenne Primes To Sq Triangular Nos.

Conjecture, For p>2, the [tex]2^{(p-1)}[/tex] th square triangular number is divisible by [tex]M_{p}[/tex] if and only if [tex]M_{p}[/tex] is prime. I checked this for 2<p<27. For instance the first four square triangular numbers are 0,1,36 and 1225 and the fourth is divisible by .
PS In fact it appears that if is prime then for any starting integers [tex]S_{1}[/tex] and [tex]S_2[/tex] having the recursive relation ,
[tex]S_{n} = 6*S_{n-1} - S_{n-2}[/tex] the following congruence holds:
[tex]S_{2^{p-1}} = S_{1} \mod M_{p}[/tex]. There is prize money lurking here for those who are interested.